We investigate the computational complexity of deciding whether k cops can capture a robber on a graph G. Goldstein and Reingold (1995) [8] conjectured that the problem is EXPTIME-complete when both...

For any integer k≥0, let ξk be the supremum in (1,2] such that the flow polynomial F(G,λ) has no real roots in (1,ξk) for all graphs G with at most k vertices of degrees larger than 3. We prove that...

A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a...

We prove that every 3-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Ω(n1/3log2⁡n) which uses at most two colors, and this bound is tight...

Robertson and the second author [7] proved in 1986 that for all h there exists f(h) such that for every h-vertex simple planar graph H, every graph with no H-minor has tree-width at most f(h); but how...

Let G be a plane graph with outer cycle C, let v1,v2∈V(C) and let (L(v):v∈V(G)) be a family of sets such that |L(v1)|=|L(v2)|=2, |L(v)|≥3 for every v∈V(C)∖{v1,v2} and |L(v)|≥5 for every v∈V(G)∖V(C)....

Consider a random graph process where vertices are chosen from the interval [0,1], and edges are chosen independently at random, but so that, for a given vertex x, the probability that there is an edge...

We study the problem of determining whether an n-node graph G contains an even hole, i.e., an induced simple cycle consisting of an even number of nodes. Conforti, Cornuéjols, Kapoor, and Vušković gave...

We investigate how to find generic and globally rigid realizations of graphs in Rd based on elementary geometric observations. Our arguments lead to new proofs of a combinatorial characterization of...

Seymour and, independently, Kelmans conjectured that every 5-connected nonplanar graph contains a subdivision of K5. We prove this conjecture for graphs containing K2,3. As a consequence, the Kelmans–Seymour...

We consider the problem of k-colouring a random r-uniform hypergraph with n vertices and cn edges, where k, r, c remain constant as n→∞. Achlioptas and Naor showed that the chromatic number of a random...

We prove that, for each nonnegative integer k and each matroid N, if M is a 3-connected matroid containing N as a minor, and the branch width of M is sufficiently large, then there is a k-element set...

Given two weighted k-uniform hypergraphs G, H of order n, how much (or little) can we make them overlap by placing them on the same vertex set? If we place them at random, how concentrated is the distribution...

We investigate when limits of graphs (graphons) and permutations (permutons) are uniquely determined by finitely many densities of their substructures, i.e., when they are finitely forcible. Every permuton...

We give a structural classification of edge-signed graphs with smallest eigenvalue greater than −2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was...

We prove that a graph admits a strongly 2-connected orientation if and only if it is 4-edge-connected, and every vertex-deleted subgraph is 2-edge-connected. In particular, every 4-connected graph has...

A digraph H is infused in a digraph G if the vertices of H are mapped to vertices of G (not necessarily distinct), and the edges of H are mapped to edge-disjoint directed paths of G joining the corresponding...

We present an easy structure theorem for graphs which do not admit an immersion of the complete graph Kt. The theorem motivates the definition of a variation of tree decompositions based on edge cuts...

A regular map is a symmetric embedding of a graph (or multigraph) on some closed surface. In this paper we consider the genus spectrum for such maps on orientable surfaces, with simple underlying graph....

A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G which embeds as a quadrangulation...

We say a digraph G is a minor of a digraph H if G can be obtained from a subdigraph of H by repeatedly contracting a strongly-connected subdigraph to a vertex. Here, we show that the class of all tournaments...

We prove that, for each prime power q, there is an integer n such that if M is a 3-connected, representable matroid with a PG(n−1,q)-minor and no U2,q2+1-minor, then M is representable over GF(q). We...

We generalize the following two seminal results.(1)Thomassen's result [15] in 1983, which says that every 4-connected planar graph is Hamiltonian-connected (which generalizes the old result of Tutte...

We show that if α is a positive real number, n and ℓ are integers exceeding 1, and q is a prime power, then every simple matroid M of sufficiently large rank, with no U2,ℓ-minor, no rank-n projective...

The Erdős–Hajnal Conjecture states that for every given H there exists a constant c(H)>0 such that every graph G that does not contain H as an induced subgraph contains a clique or a stable set of size...

We consider the chromatic number of graphs with maximum degree Δ. For sufficiently large Δ, we determine the precise values of k for which the barrier to (Δ+1−k)-colourability must be a local condition,...

Let G be a K4-free graph; an edge in its complement is a K4-saturating edge if the addition of this edge to G creates a copy of K4. Erdős and Tuza conjectured that for any n-vertex K4-free graph G with...

Let H be a tournament, and let ϵ≥0 be a real number. We call ϵ an “Erdős–Hajnal coefficient” for H if there exists c>0 such that in every tournament G not containing H as a subtournament, there is a...

The blow-up of a graph is obtained by replacing every vertex with a finite collection of copies so that the copies of two vertices are adjacent if and only if the originals are. If every vertex is replaced...

A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H′ are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is...

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity...

A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k−1)-colorable. Let fk(n) denote the minimum number of edges in an n-vertex k-critical graph. We give a lower...